Quantum Computing Using Harmonic Oscillators in the Micromaser

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Quantum Computing Using Harmonic Oscillatorsin
the Micromaser

• Dr. Ben Varcoe,
• Martin Jones, Gary Wilkes
• University of Sussex
• Department of Physics and Astronomy
• Atomic, Molecular and Optical Physics group

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Qubits ? Qudits

• Qudit d-dimensional system
• computational basis
• qubit 0?, 1?
• qudit s? s 0, 1, ... , d-1
• Dimensions of Hilbert Space
• qubit ? 2n
• qudit ? dn
• d ? ? ? Continuous Variable computation

for n quantum systems
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Physical Qudits

• Harmonic Oscillators
• Position and Momentum of a particle
• Gottesman, Kitaev and Preskill PRA 64, 012310
• Amplitude and Phase of a field
• Bartlett, de Guise and Sanders PRA 65, 052316

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The SUM Gate

• USUMa, b? ? a, (ab) mod d ? addition
  modulo-d
• e.g. for d 4 a? 3? b? 1?
• USUM3, 1?d4 ? 3, (31) mod 4? 3, 0?

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Special Case d 2

• d 2
• USUM00? ? 00? USUM01? ? 01? USUM10? ?
  11? USUM11? ? 10? ? USUM ? UCN (for d
  2)
• SUM gate is a generalised CNOT gate

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The Micromaser

• Single atoms and single modes of the field
  interact via Jaynes-Cummings Hamiltonian

from http//prola.aps.org/figure/PRA/v46/i1/p567_
1/fig1
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Micromaser Basics

• Rubidium-85 excitedby three step laser to upper
  Rydberg level
• Transition between two levels is resonant with
  microwave cavity mode
• Detection of atoms provides information about
  field

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States of interest
Coherent State
Fock State
Phase State
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States of interest
Approximate Phase State
Coherent State
Fock State
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Trapping States

• n photon number in cavity field
• g atom-field coupling
• tint interaction time
• Trapping occurs when

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Generating Phase States

• Pump parameter
• Nex effective pump rate
• ? ? 1 ? spread in n is maximised
• ? tune tint and Nex to produce phase state

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Qudits in the Micromaser

• Orthogonal, non-degenerate modes in a multimode
  cavity
• Number (Fock) State
• Phase State ??
• These are conjugate like x, p

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Scaling in the Micromaser

• Nex defines maximum Fock state in the phase state
  superposition e.g. for Nex 5
• ?? a0? b 1? c 2? d 3? e 4? f
  5?
• So two qudits give dn 36 states
• Maximum Nex ? 1500
• n 2 ? (1500)2 2.25 million states!!(compared
  to 4 for qubits)

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SUM Gate in the Micromaser

• Couple two modes via non-linear Kerr media,
• e.g. a suitable atom
• Gives
• So if t ?-1, interaction is a SUM gate
• possible in the micromaser (t ? 1/g)

(n1)P3/2
(n1)S1/2
nP3/2
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Single Qudit Operations

• Arbitrary unitary transformations by injecting
  sequences of appropriate atoms
• linear displacement of cavity mode
• squeezing of the field state
• non-linear Kerr transformations
• Fourier transform converts between Fock and phase
  eigenstates

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DiVincenzo Criteria

• ?
• ?
• ?
• ?
• ?

• A scalable physical system with
  well-characterised qudits
• The ability to initialise the qudit state
• Decoherence times much longer than the quantum
  gate operation time
• A universal set of quantum gates
• The ability to measure specific qudits

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The Future

• Desktop Micromaser Quantum PCs?
• Micromaser theory can be used in some quantum dot
  proposals
• allows miniaturisation
• better control over atoms (e.g. tint)
• very strong atom field interaction

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Microdisk Cavities

• Higher frequency (100THz vs. GHz)
• lower mode volume (?m3 vs. cm3)
• More than compensates for reduced lifetime

• Single quantum dot
• Whispering Gallery Mode
• Reduced photon lifetime

Whispering Gallery Mode
Quantum Dot
from http//www.its.caltech.edu/vahalagr/
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Summary

• Qudits offer a new and potentially more efficient
  alternative to qubits.
• The micromaser is a promising candidate for
  quantum information applications.
• Implementation of a qudit QC in the micromaser
  looks possible.
• Possibility of future incorporation into solid
  state architectures.